Percentages: Increase, Decrease, Taxes and Discounts

Percent means "out of 100": $13\%=\frac{13}{100}=0.13$. To change a percent to a decimal, divide by 100 (move the decimal point two places left). To find a percent of an amount, change the percent to a decimal and multiply: $15\%$ of $60$ is $0.15\times 60=9$. Percents can be more than 100 (a price that doubles is $200\%$ of the original) and can be less than 1 (a $0.5\%$ fee on $1000 is $5). Benchmarks help you estimate: $50\%$ is half, $25\%$ is a quarter, $10\%$ is a tenth, so $10\%$ of $80$ is $8$ and $5\%$ is half of that, $4$.

Percent change measures a change as a percent of the original (old) value: subtract to get the change, divide by the old value, multiply by 100. A positive result is a percent increase; a negative result is a percent decrease. The old value must not be zero, since you cannot divide by it. The fastest way to apply a percent change is a multiplier. An increase of $r\%$ leaves you with $100\%+r\%$ of the original, so multiply by $1+\frac{r}{100}$; a decrease of $r\%$ leaves $100\%-r\%$, so multiply by $1-\frac{r}{100}$. A 13% increase is $\times 1.13$; a 25% decrease is $\times 0.75$.

Math Antics - Calculating Percent Change

In Ontario, most purchases have 13% Harmonized Sales Tax (HST) added at the register; some items, such as basic groceries, are not taxed. The tax is a percent of the price you are actually charged, so a discount comes off first and the tax is calculated on the sale price. For the jacket: $80\times 0.75=60$, then $60\times 1.13=67.80$. Because multiplication can be done in any order, $80\times 1.13\times 0.75$ gives the same $67.80, so the order of the two multipliers does not change the total. What does matter is that the multipliers are multiplied, not the percents added: $0.75\times 1.13=0.8475$, which is a 15.25% saving overall, not 12%.

A store raises a $200 bike's price by 10%, then later runs a 10% off sale. Back to $200? No: $200\times 1.10=220$, and 10% off $220 is $22, so the price ends at $220-22=198$. The second 10% is a percent of a bigger number, so it removes more than the first one added. In multipliers: $1.10\times 0.90=0.99$, a net change of $-1\%$. The same reasoning shows that a 50% drop followed by a 50% rise leaves you at $0.5\times 1.5=0.75$ of where you started, a 25% loss. To combine successive percent changes, multiply their multipliers; never add the percents.

A receipt shows a total of $45.20 including HST. What was the price before tax? The total is $113\%$ of the pre-tax price $p$, so $1.13p=45.20$. Undo multiplication with division: $p=45.20÷1.13=40.00$. The tax was $45.20-40.00=5.20$ dollars. Undo a discount the same way: if a 20% off sale price is $64, then $0.80p=64$ and $p=64÷0.80=80$. Going forward you multiply by the multiplier; going backward you divide by it.